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Solution - Absolute value equations

Exact form:

x = - \frac{29}{2}, - \frac{3}{14}

x=-\frac{29}{2} , -\frac{3}{14}

Mixed number form:

x = - 14 \frac{1}{2}, - \frac{3}{14}

x=-14\frac{1}{2} , -\frac{3}{14}

Decimal form:

x = - 14.5, - 0.214

x=-14.5 , -0.214

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| 8 x + 16 | = | 6 x - 13 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x + 16 \| = \| 6 x - 13 \|$
$x = + y$	$(8 x + 16) = (6 x - 13)$
$x = - y$	$(8 x + 16) = - (6 x - 13)$
$+ x = y$	$(8 x + 16) = (6 x - 13)$
$- x = y$	$- (8 x + 16) = (6 x - 13)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| 8 x + 16 \| = \| 6 x - 13 \|$
$x = + y$ , $+ x = y$	$(8 x + 16) = (6 x - 13)$
$x = - y$ , $- x = y$	$(8 x + 16) = - (6 x - 13)$

2. Solve the two equations for x

9 additional steps

$(8 x + 16) = (6 x - 13)$

Subtract from both sides:

$(8 x + 16) - 6 x = (6 x - 13) - 6 x$

Group like terms:

$(8 x - 6 x) + 16 = (6 x - 13) - 6 x$

Simplify the arithmetic:

$2 x + 16 = (6 x - 13) - 6 x$

Group like terms:

$2 x + 16 = (6 x - 6 x) - 13$

Simplify the arithmetic:

$2 x + 16 = - 13$

Subtract from both sides:

$(2 x + 16) - 16 = - 13 - 16$

Simplify the arithmetic:

$2 x = - 13 - 16$

Simplify the arithmetic:

$2 x = - 29$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 29}{2}$

Simplify the fraction:

$x = \frac{- 29}{2}$

10 additional steps

$(8 x + 16) = - (6 x - 13)$

Expand the parentheses:

$(8 x + 16) = - 6 x + 13$

Add to both sides:

$(8 x + 16) + 6 x = (- 6 x + 13) + 6 x$

Group like terms:

$(8 x + 6 x) + 16 = (- 6 x + 13) + 6 x$

Simplify the arithmetic:

$14 x + 16 = (- 6 x + 13) + 6 x$

Group like terms:

$14 x + 16 = (- 6 x + 6 x) + 13$

Simplify the arithmetic:

$14 x + 16 = 13$

Subtract from both sides:

$(14 x + 16) - 16 = 13 - 16$

Simplify the arithmetic:

$14 x = 13 - 16$

Simplify the arithmetic:

$14 x = - 3$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 3}{14}$

Simplify the fraction:

$x = \frac{- 3}{14}$

3. List the solutions

$x = - \frac{29}{2}, - \frac{3}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 x + 16 |$
$y = | 6 x - 13 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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